1. Modelling Default Risk in the Trading Book: Accurate Allocation of Incremental Default Risk Charge Jan Kwiatkowski
Group Risk Management
1/1/2019

2. Summary Background on IDRC Trading book default risk models
The need for accurate allocation
Andersen, Sidenius & Basu (ASB) algorithm
Conditional Expected Loss; an allocation metric
Calculation using Bayes’ Theorem
Extensions

3. Background on IDRC
Required to find high percentile of portfolio default losses over a given time horizon.
We prefer Conditional Expected Shortfall (CES) at equivalent percentile:
The book tends to be ‘lumpy’; therefore we must include idiosyncratic effects.

4. Trading book default risk models (calibrated to a given time-horizon)
Many models use systematic risk factors and correlations

5. Conditional PDs Integrate over X (e.g. Monte Carlo or quadrature)
Require an algorithm for computing distribution of portfolio loss conditional on any X
Integrate over X (e.g. Monte Carlo or quadrature)

6. Algorithms for conditional losses
Monte Carlo
Transforms
ASB
Must keep in mind the need for allocation

7. Allocation of IDRC
Total IDRC is allocated/attributed to contributors (down to position level), and aggregated up the organisational hierarchy.
Allocation must be ‘fair’ and consistent
Especially, desks with identical positions should get the same allocation.
Using Monte Carlo for high percentiles , we are at the mercy of relatively few random numbers
Transforms not convenient for allocation

8. The ASB algorithm Discretise LGD’s as multiples of a fixed ‘Loss Unit’
ui = loss units for issuer i
Let qi = PD for issuer i (conditional on given X )
Recursively compute the distribution of the losses for portfolios consisting of the first i exposures only, for i =0, 1, 2, …., N
The method is exact modulo discretisation
Parcell (2006) shows how effects of discretisation may be mitigated
Easily extended to multiple outcomes

9. ASB implementation

10. A metric for allocation
Exactly accounts for portfolio CES

11. Bayes’ Theorem

12. Allocation methodology
We can easily calculate this by removing issuer i from the final portfolio and adding its LGD, ui, to the resulting portfolio distribution.
We use the reversal of the ASB algorithm to remove issuer i

13. Reversal of ASB
We illustrate this for a long position (ui>0); this is easily adaptable to short positions.

14. Warning This becomes unstable for qi>0 close to 1.
Can be mitigated (Parcell 2006)

15. Summary of method – Phase 1
For various systematic effects, X, use ASB to find the conditional distribution.
Integrate over X
Compute Lα, the required portfolio CES

16. Summary of method – Phase 2
For each issuer, i:
for each X
Use reverse ASB to find the distribution with i defaulted.
Compute the corresponding probability that Lα is exceeded
Multiply by uiqi/(1-α)
Integrate over X

17. Possible Extensions
The method for VaR (rather than CES) is even simpler
Multiple outcomes:
Stochastic LGDs
Rating Downgrades
Also upgrades, but requires matrix inversion
Structured products – cascade structure